1. Field of the Invention
The present invention relates to methods for characterizing existing ion channels, and designing ion channels with greater specificity for predetermined ions.
2. Background Art
Ion channels are proteins that fold to form a hole down their middle. When properly configured and installed in a cell membrane, ion channels control the movement of ions through the cell membrane. An example of this transport would be the passage of Na+, K+, Ca++, and Cl− ions from the blood stream into and out of cells. The movement of ions into and out of a cell is very important for many processes that are critically related to health and disease in living things, including people. Indeed, ion channels control an enormous range of life's functions by controlling the flow of ions and electricity in and out of cells. A sample ion channel, 100, as illustrated in the prior art is provided in FIG. 1. The port exterior to the cell is labeled 102, and the port to the interior of the cell is labeled 104. The locations of the permanent charges that are the principal influences on how the ion channel conducts ions are labeled 106.
Ion channels conduct one type of ions much better than other types of ions, and this preference for conducting one type of ion over other types of ions into or out of a cell is termed “selectivity.” The selectivity of ion channels is a crucial, indispensable part of their function.
If ion channels in living things lose their selectivity, activities critical to sustaining life cease. Thus, if the Ca++ channels of an animal or human heart were to become nonselective, or become equally selective to Na+ and Ca++, the heart could not beat and death would occur in ˜3 minutes.
Ion channels, like many other systems in biology are not perfectly formed; that is to say, they have characteristics which if changed could greatly increase their function. This characteristic of ion channels is particularly clear in the heart. A particular instance of this, the relevance of ion channels to the hearts of certain mammals is explained.
The heart is a sheet of cardiac muscle that is folded to enclose a ventricle, which is a cavity in the heart that holds the blood to be pumped. The ventricle has valves that operate to keep the blood flowing in one direction. The heart works by squeezing the blood out of the ventricle, say the left ventricle. The squeeze must start from the bottom of the ventricle (furthest away from the exit valve and exit artery called the aorta). If the squeeze starts anywhere else, contraction of the heart muscle will be futile, the ventricle will not function as a pump, and the animal or human will will quickly lose consciousness and die. So coordination of the contraction of cardiac muscle is crucial for survival.
In engineered systems of this sort, such as artificial hearts, and in the hearts of lower forms of life, coordination of this sort is done by a control system that is separate from the contractile system. Thus, most invertebrates with hearts control the heart beat with separate nervous systems, and are referred to as having neurogenic hearts. Vertebrates and humans have what are called myogenic hearts, and do not separate the control system from the heart muscle itself. Myogenic hearts use the electrical signal in the contractile tissue itself (the heart muscle) to coordinate contraction. This makes the system very labile and easy to interrupt because any difficulty the tissue has because of its contractile function immediately affects the coordination of the contraction that allows the ventricle to pump blood. Here is where channels come in. If the calcium channels responsible for (a main part of) the coordination of the heart beat are at all disrupted, the heart no longer pumps blood and the animal or person dies. If the calcium flowing through a calcium channel could be increased, the heart would be much less sensitive to this kind of disruption. Thus limitations on the current flow through a calcium channel of the heart is one example of a technical deficiency of the heart. This is just one example. There are many others which physicians and pharmacologists discover every day, unfortunately.
The structures of ion channels are currently identified using X-ray crystallography to measure the positions of the key elements of the protein. Crystallography techniques have several shortcomings, not all of which are listed here. First, the crystallography is time consuming because it takes a long time to obtain a crystal suitable for crystallographic study. Indeed, growing crystals is an art unto itself and most of the proteins of interest in membranes have not been crystallized. Second, proteins can fold differently in different environments. Therefore, even if the ion channel is crystallized, the protein as crystallized may not be in the same environment or state as it is in the animal as it functions. Thus, the crystal structure of the ion channel may not reflect the structure in the form in which it actually functions. Third, the x-ray crystallography studies are only as good as the crystals provided, and take considerable time and resources. Whether the crystal was good enough to obtain the results desired is often not known until after the study is conducted. Fourth, the expertise needed to conduct the x-ray crystallography studies is sufficiently different from the manufacture and design of the channels that it is rare for the same worker to be able to have the complete skill set, and thus workers in the field are dependent on the technical skills of a crystallographer.
The manufacture of ion channels is now sufficiently understood such that if channels can be designed to specification they can be built using the well developed techniques of molecular engineering, e.g., by site directed mutagenesis. One example is given in U.S. Pat. No. 6,979,724 to to Lerman et al. which relates to calcium channel compositions and methods of making and using them. In particular, the Lerman disclosure relates to calcium channel alpha2delta (α2δ) subunits and nucleic acid sequences encoding them. A review of ion channel manufacturing techniques is provided by [130], which should be available to the general public shortly. However, there are present shortcomings in the ability to understand how the structure of an ion channel dictates its function such that the technical ability to make an ion channel is not sufficient to solve problems in the field.
The technical limitations of present design methods are simple to state but hard to remedy. Generally, existing design methods rely on exhaustive trial and error experimentation of the highest quality, energy, and imagination to check reasonable guesses. This approach to design is inefficient and rarely works in complex systems.
There are literally thousands of laboratories doing experiments on ion channels every day all over the world. Almost all of this work is done without theoretical guidance, or rationale, using the trial and error methods of exploration and discovery of traditional biology. Such trial and error methods are essential for learning “the lay of the land”, for describing the systems and their components, but they are very inefficient for design. If it was possible to replace trial and error methods with any systematic design tool, the efficiency of thousands of laboratories would be dramatically increased, from nearly zero (which is the efficiency when design is unsuccessful) to a reasonable number.
More recent attempts at design have used simulations of atomic motions, calculations of the movement of every atom of a protein. Such calculations, despite heroic efforts and enormous computers, are unlikely to succeed because biological function occurs on millisecond time scales or slower, and atomic motions occur on 0.0000000000000005 sec time scales (femto seconds and faster). Biological function occurs (in many cases, e.g., in the channels of the heart) only when certain chemicals are present in definite concentrations. Some of these chemicals must be present in micromolar concentration. Thus, atomic scale simulations of these chemicals must include enormous numbers of atoms for many seconds, with motions on the time scale of a femtosecond being resolved. Thus, at this time, atomic scale simulations serve as metaphors and inspiration for design, but not as specific quantitative design methods. While there have been some attempts to use reduced models of channel function that do not include all atomic motions, they have not succeeded in designing a channel of a desired selectivity or in discovering the structure of an ion channel from data taken from the operation of a channel in vivo or in vitro.
Presently, the design theory of ion channels generally analyzes ion channels by linking a model for the electric field in and near the ion channel with a model for ion transport in and near the channel. Generally, such studies are currently conducted using what are termed “one dimensional models,” which model an ion channel as a line with charges placed along it in fixed locations. These placed charges are often referred to as “fixed charge” or “stationary charge” to distinguish these charges from the charges of the ions that are moving around and through the channel. The most commonly used models for the ion transport in the vicinity of the channel are built from either the Poisson-Boltzman equations or the Poisson-Nernst-Planck (PNP) equations. Other models used include, but are not limited to, simulations of Brownian motion, simulations of Brownian motion with Poisson equation, and transport Monte Carlo. Viewed in terms of physical chemistry, the models attempt to describe energetics of an ion moving through an ion channel. Viewed mathematically, these equations form what is called a system of non-linear differential equations.
In searching for ion channels with a particular selectivity, different approaches can be taken. One way of solving the selectivity problem would be to hypothesize an ion channel and ask how it might perform for different ions. Efforts to design ion channels by solving ensembles of possible ion channels in the hope of finding suitable structures have not been productive, as illustrated by references [93]-[128].
The ion channel scientists have yet to predetermine a particular selectivity for an ion channel, and then successfully attempt to determine what a channel that would have that selectivity would look like. Many kinds of scientists understand that problems can often be viewed or approached in more than one way depending on the type of information that is available and the type of solution that is desired. Thus, mathematicians know that one may know a cause or stimulus and wish to predict what will happen, or one may be observing effects and wish to infer the cause.
When searching for causes of observed or desired effects the problems are termed “inverse problems” which are likely to be difficult to solve. Two problems are called inverse to each other if the formulation of one problem involves the solution of the other one. These two problems then are separated into a direct problem and an inverse problem. At first sight, it might seem arbitrary which of these problems is called the direct and which one the inverse problem but this arbitrariness is more apparent than real. The problems have quite distinct properties and can be distinguished based on those properties.
Usually, the direct problem is the more “classical” one, in that it usually has a single, obtainable solution, which is termed “well-posed.” According to Hadamard, a mathematical problem is called well-posed if
for all admissible data, a solution exists,
for all admissible data, the solution is unique, and
the solution depends continuously on the data.
Much of the mathematical theory of partial differential equation deals with the question what “admissible data” means and in which sense “solution” is to be understood for specific classes of partial differential equations.
The direct problem usually is to predict the evolution of the studied system (described e.g. by a partial differential equation) from knowledge of its present state and the governing physical laws including information on all physically relevant parameters including boundary conditions and initial conditions. Boundary conditions are parameters that describe the behavior of the physical system or set of equations at the edges of a simulation region. Conditions imposed at the starting time for a problem where the conditions change over time are called initial conditions.
Those of ordinary skill in the art will appreciate that boundary conditions and initial conditions are much more important than they may seem at first to the uninitiated. Boundary conditions and initial conditions describe what is put into the system and what comes out of it. They describe the flow of energy, matter, electric charge, et cetera that are forced to enter and leave the system. Boundary conditions are fully as important as the system itself in determining the overall properties of a practical system. Indeed, there are many engineering systems that are designed to have specific inputs and outputs (i.e., initial and boundary conditions) only. That is to say, there are many engineering systems designed so the user does not need to be concerned what is inside the “black box” (i.e., inside the system) but only needs to be concerned with the inputs and outputs (i.e., boundary and initial conditions). Thus, in electronics, a well designed amplifier has a simple relation between input and output (called gain) and the user does not have to worry if the amplifier uses field effect transistors, bipolar transistors, or even old fashioned tubes to make that gain.
If the number and type of boundary/initial conditions is correct and the parameters are sufficiently smooth, then the direct problem is almost always well-posed and therefore easier to solve than the related inverse problem. Indeed, if the problem is not amenable to classical methods of solution (when given correct boundary/initial conditions, and reasonably smooth parameters), most scientists would conclude the theory and problem were mistaken and should be abandoned. Thus, it is characteristic of established models that their forward problems are well posed.
There are also intrinsic mathematical properties that permit one to decide which of the problems is called the “inverse” one, namely the fact that the inverse problem is usually “ill-posed.”
If one of Hadamard's conditions for terming a problem well-posed is violated, then the problem is called ill-posed. For an ill-posed problem neither the existence nor the uniqueness of a solution to an inverse problem is guaranteed. A unique solution necessarily denies the problem solver the ability to select which properties to favor in a solution. There is only one solution! However, ill-posed problems may lack any solution, or solutions may exist but are not unique (which is to say there may be more than one answer), and/or (unique or non-unique) solutions are not stable with respect to noise, modeling errors or other, even numerical, inaccuracies.
The process of bringing stability back to these problems is termed regularization. Often, regularization is done by imbedding an ill-posed problem into a collection of well-posed problems depending on some parameter, where the original ill-posed problem is a limiting case of this family of well-posed problems with respect to this parameter.
Non-uniqueness is sometimes an advantage, because non-uniqueness can allow a choice among several strategies all of which achieve a desired effect. The non-uniqueness of the solution can be advantageous because one strategy might have better properties than another. When solving design problems there is a substantial value to having a choice of solutions because that allows the problem solver to choose from different possible designs based on practical advantages not included in the mathematical model itself. This is in contrast to an identification problem where having a choice of solutions means that the identification is ambiguous.
In the case of designing ion channels it is advantageous to look for values of parameters (possibly fulfilling additional constraints) that achieve certain design goals (like selectivity in ion channel design). In contrast, in an identification problem, one wants to infer (‘identify’) values of parameters from indirect measurements, i.e., parameters are estimated not from direct measurements of the parameters but from measurements of other quantities from which estimates of the parameters are made. These other quantities appear in the mathematics as quantities in the output of the forward problem (and its boundary conditions). The inverse problem is used to estimate these parameters from measurements of the output under some conditions or other, or from multiple measurements of the output under a set of conditions (to give more information and reduce sensitivity to, for example, mistakes and noise). Here, in solving this inverse problem, uniqueness (“identifiability”) is of great importance.
Uniqueness questions are dealt with explicitly in one part of the mathematical literature of inverse problems, but as soon as one wants to compute solutions of inverse problems, one almost always has to deal with the issue of (in)stability: In practical applications, one never has exact data, but only data perturbed by noise produced by systematic or statistical errors in the measurements or produced by errors in the mathematical model itself. Models are often only a representation of reality with limited accuracy. Even if the random and/or systematic deviation from the data is small, or the error in the model is small, algorithms developed for well-posed problems will fail if they do not address the instability in the overall process of estimation of parameters. If they do not address the instability of the inverse problem due to a violation of the third Hadamard condition, data as well as round-off errors can then be amplified by an arbitrarily large factor (depending on error characteristics) arising from this lack of continuous dependence, the violation of the third Hadamard condition. This is the main effect of the lack of continuous dependence. One can quantify this effect, thus classifying ill-posed problems roughly into mildly and severely ill-posed problems depending on how strong the error amplification is. This classification has to do with the rate at which the spectrum (for a linear problem, or for the linearization of a nonlinear problem) tends to 0, see, e.g., [23].
Important and well studied classes of inverse problems which provide technical solutions to technical problems are, e.g.
(Computerized) tomography (cf. [56]), which involves the reconstruction of a function, usually a density distribution, from values of its line integrals and is important both in medical applications and in nondestructive testing [28]. Mathematically, this is connected with the inversion of the Radon transform.
Inverse scattering (cf. [17], [65]), where one wants to reconstruct an obstacle or an inhomogeneity from waves scattered by those. This is a special case of shape reconstruction and closely connected to shape optimization [41]: while in the latter, one wants to construct a shape such that some outcome is optimized, i.e., one wants to reach a desired effect, in the former, one wants to determine a shape from measurements, i.e., one is looking for the cause for an observed effect. Here, uniqueness is a basic question, since one wants to know if the shape (or anything else in some other kind of inverse problem) can be determined uniquely from the data (“identifiability”), while in a (shape) optimization problem, it might even be advantageous if one has several possibilities to reach the desired aim, so that one does not care about uniqueness there.
Inverse heat conduction problems like solving a heat equation backwards in time or “sideways” (i.e., with Cauchy data on a part of the boundary) (cf. [30]).
Geophysical inverse problems like determining a spatially varying density distribution in the earth from gravity measurements (cf. [27]).
Inverse problems in imaging like deblurring and denoising (cf. [14, 55, 60])
Identification of parameters in (partial) differential equations from interior or boundary measurements of the solution (cf. [3], [46]), the latter case appearing e.g., in impedance tomography (cf. [45]). If the parameter is piecewise constant and one is mainly interested in the location where it jumps, this can also be interpreted as a shape reconstruction problem.
Some common features of inverse problems that provide technical solutions to technical problems are problems such as amplification of high-frequency errors, a need to use one or both of a priori information and regularization to restore stability, errors of differing natures that require separate treatment, and intrinsic information loss even if one does everything in the mathematically best way. Examples of errors requiring different treatment are errors of approximation, which are how closely the model is hewing to the actual system, and the propagation of data error, wherein errors in earlier calculations cause greater errors in later calculations.
Detailed references for these and many more classes of inverse problems can be found e.g., in [23], [20], [22], [37], [53], [50], [44], [16]. In order to overcome these instabilities and design solution techniques for inverse problems which are robust (i.e., stable with respect to data and numerical errors), one has to design and use regularization methods, which in general terms replace an ill-posed problem by a family of neighboring well-posed problems.
It would be desirable to be able to control the structure and selectivity of ion channels, and even more desirable to be able to reliably design ion channels with specifically predetermined selectivity. More desirably, such methods would not use trial and error approaches that require solving ensembles of possible ion channels in the hope of fortuitously finding the desired result. This would be especially advantageous where existing ion channels are less than optimal for the function that they perform. This would be of greatest importance if the technology were able to characterize and design the selectivity of ion channel functions that are important to the life and health of animals, including humans.
Further, if the methods could pioneer the identification and design of selectivity through in vitro and in vivo experiments, the technical horizons of laboratory work in the field would be significantly broadened accelerating not only design, but also manufacture and testing. Similarly, if the technology that solved the problems of the identification of structure and design of selectivity could be applied to a broad range of models, the ability of theoretical workers to contribute to the solution of existing technical shortcomings of existing ion channels would be similarly amplified.